Bounded Symbiosis and Upwards Reflection
BBounded Symbiosis and Upwards Reflection
Lorenzo Galeotti , Yurii Khomskii , Jouko V¨a¨an¨anen
July 30, 2020
Abstract
In [2], Bagaria and V¨a¨an¨anen developed a framework for studying thelarge cardinal strength of downwards
L¨owenheim-Skolem theorems andrelated set theoretic reflection properties. The main tool was the notionof symbiosis , originally introduced by the third author in [12, 13].
Symbiosis provides a way of relating model theoretic properties ofstrong logics to definability in set theory. In this paper we continue thesystematic investigation of symbiosis and apply it to upwards
L¨owenheim-Skolem theorems and reflection principles. To achieve this, we need toadapt the notion of symbiosis to a new form, called bounded symbiosis .As one easy application, we obtain upper and lower bounds for the largecardinal strength of upwards L¨owenheim-Skolem-type principles for sec-ond order logic.
Mathematicians have two ways of characterizing a class C of mathematical struc-tures: definining the class in set theory, or axiomatizing the class by sentencesin logic. Symbolically:1. Φ( A ), where Φ is a formula in the language of set theory, vs.2. A | = ϕ , where ϕ is a sentence in some logic.In general, set theory is much more powerful than first order logic.However, by restricting the allowed complexity of Φ on one hand, whileconsidering extensions of first-order logic on the other hand, one gets a more in-teresting picture. Symbiosis aims to capture an equivalence in strength betweenset-theoretic definability and model-theoretic axiomatisability. One applicationof this is connecting properties of some strong logic L ∗ to specific set-theoreticprinciples (often expressed in terms of large cardinals). Symbiosis was firstintroduced by the third author in [13], and studied further in [9, 1, 2]. Amsterdam University College, Postbus 94160, 1090 GD Amsterdam, The Netherlands Institute for Logic, Language and Computation, Universiteit van Amsterdam, Postbus94242, 1090 GE Amsterdam, The Netherlands Universit¨at Hamburg, Bundesstraße 55, 20146 Hamburg This author has received funding from the European Unions Horizon 2020 research andinnovation programme under the Marie Sk(cid:32)lodowska-Curie grant agreement No 706219 (REG-PROP). Department of Mathematics and Statistics, University of Helsinki, Finland The author would like to thank the Academy of Finland, grant no.: 322795 a r X i v : . [ m a t h . L O ] J u l f A is a structure and φ a first-order formula, then the statement “ A | = φ ”is ∆ in set theory. Therefore every first-order axiomatizable class of structures,i.e., every class of the form Mod( φ ) = {A : A | = φ } , is ∆ -definable.The converse does not hold: for example, the class of all well-ordered struc-tures is easily seen to be ∆ -definable but not first-order axiomatizable. So it isnatural to look for a logic L ∗ extending first order logic, with the property thatevery ∆ -definable class would be axiomatizable by an L ∗ -sentenceConsider the logic L I = L ωω (I) obtained from first order logic L ωω by addingthe H¨artig quantifier
I, defined by
A | = I xy φ ( x ) ψ ( y ) iff |{ a ∈ A : A | = φ [ a ] }| = |{ b ∈ A : A | = ψ [ b ] }| and consider its closure under the so-called ∆-operator (Definition 3.1). Wethen obtain a logic, which we will call ∆( L I ), such that every ∆ -definable class,if closed under isomorphisms, is ∆( L I )-axiomatisable (see Proposition 3.5 or [13,Example 2.3]).However, ∆( L I )-axiomatisability is now too strong to be “symbiotic” with∆ -definability: the class { ( A, P ) : |{ x ∈ A : P ( x ) }| = |{ x ∈ A : ¬ P ( x ) }|} is not ∆ (it is not absolute), but it is axiomatisable in L I by the sentenceI xy ( P ( x ))( ¬ P ( y )) . One can observe that all ∆( L I )-axiomatisable classes are ∆ -definable, butonce more, there are ∆ -definable classes that are not ∆( L I )-axiomatisable (seeFigure 1). ∆ (cid:79) (cid:79) ∆( L I ) (cid:79) (cid:79) (cid:105) (cid:105) ∆ (cid:53) (cid:53) L ωω (cid:105) (cid:105) Set Theory LogicFigure 1: Set-theoretic definability vs. axiomatization in a logicInteresting symbiosis relationships take place for complexity levels ∆ ( R ),for fixed predicates R . In this paper, we will focus on Π predicates R , so thecomplexity levels will lie below ∆ . Many such relations have been established Usually the symbol used here is a simple ∆, but in this paper we choose the symbol∆, and similarly Σ, in order to easily distinguish the model-theoretic notions from the L´evycomplexity of formulas in the language of set theory, i.e., Σ n and ∆ n formulas.
2n [13, 2]. To name some prominent examples, let L be second order logicwith full semantics, and let L WF be the logic obtained from L ωω by adding thegeneralized quantifier WF defined by A | = WF xy φ ( x, y ) iff { ( x, y ) ∈ A × A : A | = φ ( x, y ) } is well-founded.Furthermore, let Cd( x ) be the Π predicate “ x is a cardinal”, and let PwSt( x, y )be the Π predicate “ y = ℘ ( x )”. Then we have the symbiosis relationshipsdepicted in Figure 2 (see Propositions 3.4, 3.5 and 3.6).∆ (PwSt) (cid:79) (cid:79) ∆( L ) (cid:79) (cid:79) ∆ (Cd) ∆( L I )∆ ∆( L WF ) L ωω Set Theory LogicFigure 2: Symbiosis relationsAs an application of symbiosis , Bagaria and V¨a¨an¨anen [2] considered thefollowing principles:
Definition 1.1.
The downward L¨owenheim-Skolem-Tarski number
LST ( L ∗ ) isthe smallest cardinal κ such that for all φ ∈ L ∗ , if A | = L ∗ φ then there exists asubstructure B ⊆ A such that | B | ≤ κ and B | = L ∗ φ . If such a κ does not exist, LST ( L ∗ ) is undefined. Definition 1.2.
Let R be a predicate in the language of set theory. The struc-tural reflection number SR ( R ) is the smallest cardinal κ such that for everyΣ ( R )-definable class K of models in a fixed vocabulary, for every A ∈ K thereexists
B ∈ K with | B | ≤ κ and a first-order elementary embedding e : B (cid:52) A .If such a κ does not exists, SR ( R ) is undefined. Theorem 1.3 (Bagaria & V¨a¨an¨anen [2]) . Suppose L ∗ and R are symbiotic.Then LST ( L ∗ ) = κ if and only if SR ( R ) = κ .Proof. See [2, Theorem 6].Theorem 1.3 links a meta-logical property of a strong logic to a reflectionprinciple in set theory. Depending on the predicate R , the principle SR ( R ) has avarying degree of large cardinal strength. In fact, Definition 1.2 may be regardedas a kind of Vopˇenka principle, restricted to classes of limited complexity. See [1, Sections 3, 4] for more on the connection between SR ( R ) and Vopˇenka-typeprinciples. SR ( R ) and LST ( L ∗ ) for various symbiotic pairs.In this paper we continue the work of Bagaria and V¨a¨an¨anen by developing aframework for the study of upward L¨owenheim-Skolem and reflection principles.These principles are also interesting because they are closely related to the compactness of the strong logic, although the two notions are not equivalent, andin this paper we do not consider compactness explicitly. The main innovation ofthe current work is that, in order to deal with upwards rather than downwardsreflection, we need to adapt the notion of symbiosis .The paper is organized as follows: in Section 2 we introduce the necessaryterminology and some background and in Section 3 we present the notion of symbiosis . In Section 4 we introduce bounded symbiosis . Section 5 is devoted toexamples of bounded symbiosis , and in Section 6 we prove the main theorem,showing that under appropriate conditions the upward
L¨owenheim-Skolem num-ber corresponds to a suitable upwards set-theoretic reflection principle. Finally,in Section 7 we apply our results to compute upper and lower bounds for theupward L¨owenheim-Skolem number of second order logic and the correspondingreflection principle, noting that this also provides an upper bound for all otherΠ predicates.This paper contains research carried out by the first author as part of hisPhD Dissertation. Some details have been left out of the paper for the sake ofeasier readability and navigation. The interested reader may find these detailsin [5, Chapter 6]. We assume that the reader is familiar with standard set theoretic and modeltheoretic notation and terminology. We will consider abstract logics L ∗ , withoutproviding a precise definition for what counts as a “logic”. Typical examplesare infinitary logics L κλ , full second-order logic L , and various extensions offirst-order logic by generalized quantifiers . For a more detailed analysis we referthe reader to [5, Chapter 6] and [3]. Here we only want to stress two importantpoints.First, we will generally work with many-sorted languages, using the symbols s , s , . . . to denote sorts . In this setting, a domain may be a collection ofdomains (one for each sort), and all constant, relation and function symbolsmust have their sort specified in advance. This is a matter of convenience,since many-sorted logic can be simulated by standard single-sorted logic byintroducing additional predicate symbols. The following definition is essentialfor what follows. Definition 2.1.
Suppose that τ ⊆ τ (cid:48) are many-sorted vocabularies and that M is a τ (cid:48) -structure. The reduct (or projection ) of M to τ , denoted by M (cid:22) τ , isthe structure whose domains are those domains in the sorts available in τ , andthe interpretations of all symbols not in τ are ignored.4o a reduct M (cid:22) τ can have a smaller domain, and a smaller cardinality, thanthe original model M .Secondly, we should note that one needs to be careful with the syntax of agiven logic, because an unrestricted use of syntax may give rise to some unde-sirable effects. Consider, for example, an arbitrary set X ⊆ ω , and a vocabulary τ which has ω -many relation symbols { R i : i < ω } , such that the arity of R i is 1 if i ∈ X and 2 if i / ∈ X . The information about arities of relation symbolsmust be encoded in the vocabulary τ . Therefore, the set X can be computedfrom τ .In infinitary logic, we can encode a set X ⊆ ω even with finite vocabularies.Let φ , φ , . . . be a recursive enumeration of L ωω -sentences in some fixed τ , andconsider the L ω ω -sentence Φ := (cid:86) n ∈ X φ n . Then X can be computed from Φ. Definition 2.2.
Let L ∗ be a logic. The dependence number of L ∗ , dep( L ∗ ), isthe least λ such that for any vocabulary τ and any L ∗ -formula φ in τ , there isa sub-vocabulary σ ⊆ τ such that | σ | < λ and φ only uses symbols in σ . If sucha number does not exist, dep( L ∗ ) is undefined. Definition 2.3.
We say that a logic L ∗ has ∆ -definable syntax if every L ∗ -formula is ∆ -definable in set theory (as a syntactic object), possibly with thevocabulary of φ as parameter.In our main Theorem 6.3, we will restrict attention to logics with a ∆ -definable syntax and dep( L ∗ ) = ω . Note that this includes all finitary logicsobtained by adding finitely many generalized quantifiers to first- or second-orderlogic.We end this section by defining a version of the upward L¨owenheim-Skolemnumber for abstract logics. Definition 2.4 (Upward L¨owenheim-Skolem number) . Let L ∗ be a logic.1. The upward L¨owenheim-Skolem number of L ∗ for <λ -vocabularies , de-noted by ULST λ ( L ∗ ), is the smallest cardinal κ such thatfor every vocabulary τ with | τ | < λ and every φ in L ∗ [ τ ], if thereis a model A | = φ with |A| ≥ κ , then for every κ (cid:48) > κ , there isa model B | = φ such that |B| ≥ κ (cid:48) and A is a substructure of B .As usual, if there is no such cardinal then ULST λ ( L ∗ ) is undefined.2. The upward L¨owenheim-Skolem number of L ∗ , denoted by ULST ∞ ( L ∗ )is the smallest cardinal κ such that ULST λ ( L ∗ ) ≤ κ for all cardinals λ .Again, if there is no such cardinal then ULST ∞ ( L ∗ ) is undefined.Notice that when dep( L ∗ ) = λ , then ULST λ ( L ∗ ) = κ implies ULST ∞ ( L ∗ ) = κ . In general, ULST ∞ ( L ∗ ) may fail to be defined even if all ULST λ ( L ∗ ) = κ aredefined.Recall also that the Hanf-number of a logic is defined analogously to Def-inition 2.4 but without the assumption that A is a substructure of B . Thisadditional assumption is rather crucial: it is easy to see that if the dependencenumber of a logic is defined, then the Hanf number is also defined (see [3, The-orem 6.4.1]). However, as we shall see in Section 7, the existence of upwardL¨owenheim-Skolem numbers in the sense of Definition 2.4, even for logics withdependence number ω , implies the existence of large cardinals.5 Symbiosis
Symbiosis was introduced by the third author in [13]. To motivate its definition,let L ∗ be a logic and R a predicate in set theory. The aim is to establish an equal-ity in strength between L ∗ -axiomatizability and ∆ ( R )-definability. One direc-tion should be the statement “the satisfaction relation | = L ∗ is ∆ ( R )-definable”,or, equivalently, “every L ∗ -axiomatizable class of structures is ∆ ( R )-definable.”The converse direction should say, roughly speaking, that every ∆ ( R )-definable class is L ∗ -axiomatizable. This cannot literally work, because L ∗ -model classes are closed under isomorphisms whereas this is not necessarilytrue for arbitrary ∆ ( R ) classes. Therefore we try the approach “every ∆ ( R )-definable class closed under isomorphisms is L ∗ -axiomatizable.”Unfortunately, this does not always work: symbiosis can only be estab-lished for logics that are closed under the ∆-operation. This operation closesthe logic under operations which are in a sense “simple” but not as simple asmere conjunction, negation, and whatever operations the logic has. In orderto define the property of a structure A being in a model class based merely onthe knowledge that the class is ∆ ( R ), the only way seems to be to use themeans of the logic to build a piece of the set theoretic universe around A , andwork in the small universe. The ∆-operation is then used to eliminate the extrasymbols used to build the small universe. See [2, 13, 11] for more details on the∆-operation and its use. Definition 3.1.
Let L ∗ be a logic and let τ a fixed vocabulary. A class K of τ -structures is Σ( L ∗ ) -axiomatisable if there exists φ in some extended vocabulary τ (cid:48) ⊇ τ such that K = {A : ∃B ( B | = φ and A = B (cid:22) τ ) } . We say that K is the projection of the class Mod( φ ) to τ .A class K is Π( L ∗ ) -axiomatisable if the complement of K (i.e., the class of τ -structures not in K ) is Σ( L ∗ )-axiomatizable, and ∆( L ∗ )-axiomatizable if it isboth Σ( L ∗ ) and Π( L ∗ )-axiomatizable.Note that, if τ (cid:48) has more sorts than τ , then the structures B can be largerthan their reducts A = B (cid:22) τ .Since ∆( L ∗ )-axiomatizable classes are closed under unions, intersections,complements and projections, one could consider ∆( L ∗ ) itself as an abstractlogic, whose model classes are exactly the ∆( L ∗ )-axiomatizable classes. In gen-eral, ∆( L ∗ ) is a non-trivial extension of L ∗ . However, for first-order logic, andin general any logic satisfying the the Craig Interpolation Theorem , two notionscoincide, see [11, Lemma 2.7].
Definition 3.2 ( Symbiosis ) . Let L ∗ be a logic and R a predicate in the lan-guage of set theory. Then we say that L ∗ and R are symbiotic if:(1) the relation | = L ∗ is ∆ ( R )-definable, and(2) for every finite vocabulary τ , every ∆ ( R )-definable class of τ -structuresclosed under isomorphisms is ∆( L ∗ )-axiomatisable.6n [2], symbiosis was established for many logic-predicate pairs, among themthe ones mentioned in the introduction.In practice, there is an equivalent condition to (2) which is easier to bothverify and to apply. Let R be an n -ary predicate in the language of set the-ory. We say that a transitive model of set theory M is R -correct if for all m , . . . , m n ∈ M we have M | = R ( m , . . . , m n ) iff R ( m , . . . , m n ). Lemma 3.3 (V¨a¨an¨anen [13]) . For any predicate R and logic L ∗ , the followingare equivalent:(a) For every finite τ , every ∆ ( R ) class of τ -structures closed under isomor-phisms is ∆( L ∗ ) -axiomatisable.(b) The class Q R := {A : A is isomorphic to a transitive R -correct ∈ -model } is ∆( L ∗ ) -axiomatisable.Proof. See [13], or a simpler version of Theorem 4.9.For completeness, and to illustrate how proofs of symbiosis typically workin view of the results in the next section, we will now sketch proofs of someparadigmatic examples (see Section 1 for the definitions).
ZFC −∗ refers to asufficiently large fragment of ZFC − Power Set.
Proposition 3.4 (V¨a¨an¨anen [13]) . L WF and ∅ (no predicates) are symbiotic.Proof. (1) Since the statement “( A, E ) is well-founded” for sets A is ∆ andtherefore absolute for transitive models, “ A | = L WF φ ” is also absolute fortransitive models. Then A | = L WF φ iff ∃ M ( M transitive ∧ M | = ZFC −∗ ∧ A ∈ M ∧ M | = ( A | = L WF φ ))iff ∀ M (( M transitive ∧ M | = ZFC −∗ ∧ A ∈ M ) → M | = ( A | = L WF φ )).This gives a ∆ -definition.(2) There are no predicates so Q ∅ = { ( A, E ) : (
A, E ) is isomorphic to a tran-sitive ∈ -model } . But ( A, E ) is isomorphic to a transitive ∈ -model iff E iswell-founded and extensional. Therefore ( A, E ) ∈ Q ∅ iff( A, E ) | = Ext ∧ WF xy ( xEy )which is an L WF -sentence. Thus Q ∅ is L WF -axiomatizable and thereforealso ∆( L WF )-axiomatizable. Proposition 3.5 (V¨a¨an¨anen [13]) . L I and Cd are symbiotic.Proof. (1) It is easy to see that “ A | = L I φ ” is absolute for models of set theory whichare Cd-correct. Therefore A | = L I φ iff ∃ M ( M trans. and Cd-correct ∧ M | = ZFC −∗ ∧ A ∈ M ∧ M | =( A | = L I φ ))iff ∀ M (( M trans. and Cd-correct ∧ M | = ZFC −∗ ∧ A ∈ M ) → M | = ( A | = L I φ ))Note that “ M is Cd-correct” is the statement ∀ x ∈ M (( M | = Cd( x )) ↔ Cd( x )) which is ∆ (Cd). Thus the above is a ∆ (Cd) formula.72) We need to check that Q Cd = { ( A, E ) : (
A, E ) is isomorphic to a transitiveCd-correct ∈ -model } is ∆( L I ). We have:( A, E ) ∈ Q Cd iff(a) E is wellfounded(b) ( A, E ) | = Ext (c) (
A, E ) | = L I “ ∀ α (Cd( α ) → ∀ x ∈ α ¬ Iyz ( y ∈ x )( y ∈ α ))” (written in E instead of ∈ ).Conditions (b) and (c) are L I -sentences, so it remains to show that (a) is∆( L I ).First, we add a new unary predicate symbol P and consider the sentence“ P has no E -least element”, i.e., φ ≡ ∀ x ( P ( x ) → ∃ y ( P ( y ) ∧ yEx )) . Clearly the class of all models (
A, E ) such that E is not well-founded is theprojection of Mod( φ ) to { E } . Therefore (a) is Π( L I ).To show that (a) is also Σ( L I ) we use a trick due to Per Lindstr¨om [7]:( A, E ) is well-founded if and only if we can associate sets X a to every a ∈ A in such a way that aEb → | X a | < | X b | . Let the original sort be called s , extend the language with a second sort s , add a new binary relationsymbol R from s to s , and consider the sentence φ ≡ ∀ a ∀ b ( aEb → ( ∀ x ( R ( a, x ) → R ( b, x )) ∧ ¬ IyzR ( a, y ) R ( b, z )))where we have used ∀ and ∀ to denote quantification over the two sorts.Now we can easily see that if A = ( A, X, E A , R A ) | = φ , then the sets X a := { x ∈ X : R A ( a, x ) } are exactly as required, hence E M is well-founded. Conversely, if ( A, E A ) is well-founded, let rk E : A → Ord bethe rank function induced by E A , let X := ℵ sup a ∈ A { rk E ( a )+1 } , and define R A ⊆ A × X by R A ( a, α ) ⇔ α < ℵ rk E ( a ) . Then ( A, X, E A , R A ) | = φ .Thus we conclude that the class of well-founded structures is the projectionof Mod( φ ) to s , completing the proof. Proposition 3.6 (V¨a¨an¨anen [13]) . L and PwSt are symbiotic.Proof. (1) As before, note that the statement “
A | = L φ ” is absolute for tran-sitive models which are PwSt-correct, so we have A | = L φ iff ∃ M ( M trans. and PwSt-correct ∧ M | = ZFC −∗ ∧ A ∈ M ∧ M | =( A | = L I φ ))iff ∀ M (( M trans. and PwSt-correct ∧ M | = ZFC −∗ ∧ A ∈ M ) → M | = ( A | = L I φ )) 82) Consider Q PwSt = { ( A, E ) : (
A, E ) is isomorphic to a PwSt-correct ∈ -model } . We have ( A, E ) ∈ Q PwSt iff E is wellfounded and extensionaland ( A, E ) | = L ( y = ℘ ( x ) ↔ Φ( x, y )) where Φ( x, y ) is the L -formulaexpressing that y is the true power set of x , written using E instead of ∈ . All of this is expressible in L . Although symbiosis is stated as a property of L ∗ , it is really a property of∆( L ∗ ). For many applications, this is irrelevant: for example, the downwards L¨owenheim-Skolem principles are all preserved by the ∆-operation. However, in[14, Theorem 4.1] it was shown that the Hanf-number may not be preserved, andthe bounded ∆-operation was introduced as a closely related operation whichstill fulfills most of the properties but, in addition, preserves Hanf-numbers. Thebounded ∆ coincides with the original ∆ in many but not all cases, see [14].If we want to apply symbiosis to upwards L¨owenheim-Skolem principles, weneed to accommodate this bounded version of the ∆-operation. Unfortunately,this also requires adapting the set-theoretic complexity classes to bounded ver-sions. This section is devoted to the definition of these concepts. We start withthe model-theoretic side. The following definition generalizes Definition 3.1 andwas first introduced in [14, p. 45].
Definition 4.1.
A class K of τ -structures is Σ B ( L ∗ ) -axiomatisable if thereexists φ in an extended vocabulary τ (cid:48) ⊇ τ such that K = {A : ∃B ( B | = φ and A = B (cid:22) τ ) } , and for all A there exists a cardinal λ A , such that for any τ (cid:48) -structure B : if B | = φ and A = B (cid:22) τ then |B| ≤ λ A .We say that K is a bounded projection of Mod( φ ). K is ∆ B ( L ∗ ) -axiomatisable if both K and its complement are Σ B ( L ∗ )-axiomatisable.In other words, K is a bounded projection of Mod( φ ) if it is a projectionand, in addition, every structure B ∈
Mod( φ ) is bounded in its cardinality bya function that depends on the respective reduct B (cid:22) τ . Note that this definitionreally only plays a role when the extended vocabulary has additional sorts, sinceotherwise the cardinalities of B (cid:22) τ and B are the same.Typical examples of bounded projection will be seen, e.g., in Propositions5.1, 5.2 and 5.6.It will be useful to define a bound given by a function from ordinals toordinals rather than models to ordinals. If we use superscripts 0 and 1 to denote first- and second-order quantification, and the re-lation symbols ∈ and ∈ to denote sets-in-sets membership and sets-in-classes membership,respectively, the sentence Φ( x, y ) can be written as follows: ∀ Z ( ∃ v ( v ∈ y ∧ ∀ w ( w ∈ v ↔ w ∈ Z )) ↔ ∀ v ( v ∈ Z → v ∈ y )) emma 4.2. Suppose K is Σ B ( L ∗ ) -axiomatisable. Then there exists φ in anextended vocabulary τ (cid:48) ⊇ τ , and a non-decreasing function h : Ord → Ord suchthat K = {A : ∃B ( B | = φ and A = B (cid:22) τ ) } and ∀B ( B | = φ → |B| ≤ h ( |B (cid:22) τ | )) . Proof.
Define h by h ( λ ) := sup { λ A : |A| ≤ λ } where each λ A is as in Definitions4.1. Since there are only set-many non-isomorphic models of any cardinality, h is well-defined.Now we move to the set-theoretic side of things, which is more involved. Inparticular, we may no longer refer to arbitrary Σ -formulas, since the witness insuch formula may be unbounded, making it impossible to establish a symbiotic relationship for bounded projective classes. So we would like to restrict attentionto formulas φ ( x ) of the form ∃ y ψ ( x, y ) but where, in addition, the (hereditary)size of at least one witness y is bounded by a function of the (hereditary) sizeof x , and this function itself can be “captured” by first-order logic. Note that asimilar concept was introduced by the third author in [13, Definition 3.1].We first need to introduce the concept of “being captured by first-orderlogic”. Definition 4.3.
A non-decreasing class function F : Card → Card is called definably bounding if the class of structures K := { ( A, B ) : | B | ≤ F ( | A | ) } (in the vocabulary with two sorts and no symbols) is Σ B ( L ωω )-axiomatisable.The intuition here is that the size of | B | (the “witness”) may be larger than | A | , but not by too much — and by exactly how much is determined by F . Forexample, the identity function F = id is definably bounding since we can alwaysextend the language with a new function symbol f between the two sorts andexpress “ f is a surjection” in L ωω . A more interesting example is the following: Example 4.4.
The function F ( κ ) = 2 κ is definably bounding. Proof.
Consider the class K := { ( A, B ) : | B | ≤ | A | } . Extend the vocabu-lary with a new relation symbol E between the two sorts in reverse order, andconsider the first-order formula φ ≡ ∀ B b, b (cid:48) ( ∀ A a ( aEb ↔ aEb (cid:48) ) → b = b (cid:48) ) , where we used the notation ∀ A and ∀ B to informally refer to quantificationover the sorts. It is easy to see that if M = ( A, B, E M ) | = φ then the map i : B → ℘ ( A ) given by i ( b ) := { a ∈ A : aE M b } is injective, so | B | ≤ | ℘ ( A ) | . Itfollows that K is the projection of Mod( φ ). The “bounded” part is immediatesince we have not added new sorts.By an additional argument (see [5, Lemma 6.23]), it is not hard to provethat if F is definably bounding, then so is any iteration F n . In particular, if wedefine the cardinal function (cid:105) n for infinite cardinals λ by setting (cid:105) ( λ ) := 2 λ and (cid:105) n +1 ( λ ) := 2 (cid:105) n ( λ ) , then each such (cid:105) n is also definably bounding. This istypically strong enough for most interesting applications.10 efinition 4.5.
1. For a set x , the H -rank of x , denoted by ρ H ( x ), is the least infinite κ suchthat x ∈ H κ + (i.e., ρ H ( κ ) = min( ℵ , | trcl( x ) | )).2. Let F be a definably bounding function. A set-theoretic formula φ ( x ) isΣ F if there exists a ∆ formula ψ ( x, y ) such that(a) ∀ x ( φ ( x ) ↔ ∃ y ψ ( x, y )), and(b) ∀ x ( φ ( x ) → ∃ y (cid:48) ( ρ H ( y (cid:48) ) ≤ F ( ρ H ( x )) ∧ ψ ( x, y (cid:48) ))A formula is Π F if its negation is Σ F , and ∆ F if it is equivalent to botha Σ F - and a Π F -formula.3. Let R be a predicate in the language of set theory. All of the above canbe generalized to Σ F ( R ), Π F ( R ) and ∆ F ( R ) in the obvious way.So, a Σ F formula is a Σ formula such that, in addition, at least one “witness” y is not too far up in terms of H -rank in relation to x itself, where by “not too farup” we mean “bounded by the definably bounding function F ”. An importantexample is the satisfaction relation of first-order logic: Remark 4.6.
The satisfaction relation | = L ωω is ∆ id1 (see [4]).This leads us to introduce a new notion of symbiosis. A similar idea alreadyappeared in [13, Definition 3.3]. Definition 4.7 ( Bounded Symbiosis ) . Let L ∗ be a logic and R a set theoreticpredicate. We say that L ∗ and R are boundedly symbiotic if(1) The relation | = L ∗ is ∆ F ( R )-definable for some definably bounding F .(2) Every ∆ F ( R )-definable class of τ -structures closed under isomorphisms is∆ B ( L ∗ )-axiomatizable (for every definably bounding F ).Just as before, condition (2) of bounded symbiosis has an equivalent formwhich is usually easier to verify and to apply. A side effect will be that in (2), wemay assume that F = id without loss of generality. First, we need the following: Lemma 4.8.
Let R be a Π predicate in set theory.1. Every H κ is R -correct.2. Let φ be a Σ F ( R ) -formula. Then for every x and every κ > F ( ρ H ( x )) , φ ( x ) is absolute (upwards and downwards) for H κ .Proof. ( R )-formulas are absolute for H κ .For 2, it suffices to prove downwards absoluteness. Let x be arbitrary andsuppose φ ( x ) holds. Then there exists y such that ρ H ( y ) ≤ F ( ρ H ( x )) < κ and ψ ( x, y ) holds, where ψ ( x, y ) is the corresponding ∆ ( R )-formula. But then y ∈ H κ and H κ | = ψ ( x, y ) by the above. It follows that H κ | = φ ( x ). Lemma 4.9.
Let L ∗ be a logic and R be a Π predicate. Then the followingare equivalent: a) Every ∆ F ( R ) class of τ -structures closed under isomorphisms is ∆ B ( L ∗ ) -axiomatisable (for every definably bounding F ).(b) Every ∆ id1 ( R ) class of τ -structures closed under isomorphisms is ∆ B ( L ∗ ) -axiomatisable.(c) The class Q R := {A : A is isomorphic to a transitive R -correct ∈ -model } is ∆ B ( L ∗ ) -axiomatisable.Proof. ( a ) ⇒ ( b ) is immediate. For ( b ) ⇒ ( c ), it is enough to prove that Q R itself is ∆ id1 ( R )-definable. We have A ∈ Q R iff ∃ M ∃ f such that1. ρ H ( M ) ≤ ρ H ( A )2. ρ H ( f ) ≤ ρ H ( A )3. M is transitive4. f : A = ( A, E ) ∼ = ( M, ∈ ) is an isomorphism5. ∀ x . . . x n ∈ M ( M | = R ( x , . . . , x n ) ↔ R ( x , . . . , x n ))Since clauses 3–5 are ∆ ( R ), this gives a Σ id1 ( R ) statement. Similarly, A / ∈ Q R iff ( A, E ) is not well-founded or ∃ M ∃ f such that1. ρ H ( M ) ≤ ρ H ( A )2. ρ H ( f ) ≤ ρ H ( A )3. M is transitive4. f : A = ( A, E ) ∼ = ( M, ∈ ) is an isomorphism5. ¬∀ x . . . x n ∈ M ( M | = R ( x , . . . , x n ) ↔ R ( x , . . . , x n ))It is easy to see that being ill-founded is Σ id1 , so the conjunction is again aΣ id1 ( R ) statement.Now we look at ( c ) ⇒ ( a ). Let K be a ∆ F ( R )-definable class over a fixedvocabulary τ which is closed under isomorphisms. Let Φ( x ) be the Σ F ( R )formula defining K . For simplicity, assume that τ consists only of one binarypredicate P and only one sort.Let τ (cid:48) be a language in two sorts s and s , with E a binary relation symbol ofsort s , G a function symbol from s to s , c a constant symbol of sort s , and P a unary predicate symbol in s (i.e., s is the original sort of τ , while s addsa “model of set theory” on the side).Let K (cid:48) be the class of all τ (cid:48) -structures M := (cid:0) M, A, E M , c M , G M , P M (cid:1) satisfying the following conditions:1. ( M, E M ) ∈ Q R , i.e., ( M, E M ) is isomorphic to a transitive R -correctmodel 12. ( M, E M ) | = ZFC −∗ M | = Φ( c )4. | M | ≤ F ( | A | ) M | = “ c = ( a, b ) and b ⊆ a × a ” (written using E instead of ∈ )6. M | = “ G is an isomorphism between ( A, P ) and ( a, b ) ( M,E ) ”. In thissentence, ( a, b ) ( M,E ) refers to the domain and binary relation on it whichis described by a and b when interpreting ∈ by E M (e.g., the domain isreally { x ∈ M : xE M a } etc. )Now we can see that conditions 2, 3, 5 and 6 are directly expressible in L ωω ,while 1 is ∆ B ( L ∗ )-axiomatisable, and hence Σ B ( L ∗ )-axiomatisable, by assump-tion. Moreover, 4 is also Σ B ( L ∗ )-axiomatisable: this follows by the definition of“definably bounding”, by Example 4.4, and the discussion following it.It remains to prove that K is a bounded projection of K (cid:48) to τ . Note that the“bounded” part is immediate due to 4. • First suppose M = (cid:0) M, A, E M , c M , G M , P M (cid:1) ∈ K (cid:48) . We want to showthat ( A, P M ) ∈ K . By 1 ( M, E M ) is isomorphic to a transitive model( M , ∈ ) which is R -correct. Let c M be the image of c M under this iso-morphism. Then ( M , ∈ ) | = Φ( c M ). Moreover, since M is R -correct andΦ is Σ ( R ), by upwards absoluteness we have Φ( c M ), i.e. c M ∈ K . By 6we have c M ∼ = “( a, b ) ( M,E ) ” ∼ = ( A, P M ). Since K is closed under isomor-phism, it follows that ( A, P M ) ∈ K . • Conversely, let A = ( A, P A ) ∈ K , i.e., Φ( A ) holds. We want to find astructure M ∈ K (cid:48) such that A = M (cid:22) τ .The first idea would be to find an H θ which is sufficiently large to reflectΦ( A ) while still being small enough to satisfy condition (4). In general,however, the transitive closure of A might be significantly larger than | A | .So we first find a model ¯ A which is isomorphic to A but whose domain issome cardinal µ . Since K is closed under isomorphisms, ¯ A is also in K ,i.e. Φ( ¯ A ) also holds.Note that in this case P ¯ A ⊆ µ × µ , in particular, trcl( ¯ A ) = trcl(( µ, P ¯ A )) ⊆ µ , so ρ H ( ¯ A ) = µ . Let θ := F ( µ ) + . By Lemma 4.8 (2) H θ | = Φ( ¯ A ).Now let M = ( H θ , A, ∈ , ¯ A , g, P A ), where g is the isomorphism between A and ¯ A . Now it is not hard to verify that all 6 conditions in the definitionof K (cid:48) are satisfied. In particular, 1 holds because of Lemma 4.8 (1) and 4because | H θ | ≤ θ = 2 F ( µ ) + ≤ F ( µ ) = 2 F ( | A | ) . Thus
M ∈ K (cid:48) as we wanted.This shows that K is a bounded projection of K (cid:48) and therefore K is Σ B ( L ∗ ).Since K is also ∆ F ( R ), the same proof works for the complement of K , showingthat K is ∆ B ( L ∗ ). Even though we made an assumption to only consider the language τ with one binaryrelation symbol for the sake of clarity, the same holds for any number of predicate or functionsymbols on a model with domain µ . Examples of Bounded Symbiosis
In general, all the pairs that are proved to be symbiotic in [2, Proposition 4]are also bounded symbiotic.
For completeness, we now show how the proofs ofPropositions 3.4, 3.5 and 3.6 can be strengthened to prove bounded symbio-sis. In particular, Proposition 5.2 is a non-trivial result since by [14, §
4] it isconsistent that ∆( L I ) (cid:54) = ∆ B ( L I ). Proposition 5.1.
The pairs L WF and ∅ are bounded symbiotic.Proof. The same proof as Proposition 3.4 works. For (1), note that we mayalways use reflection to find M such that | M | = | trcl( A ) | . This implies that | = L WF is ∆ id1 . For part (2), Q WF is actually L WF -definable, hence ∆ B ( L WF )-definable. Proposition 5.2.
The pairs L I and Cd are bounded symbiotic.Proof. Again we look at the proof of Proposition 3.5. For (1), we have thestronger equivalence:
A | = L I φ iff ∃ M ( ρ H ( M ) ≤ ρ H ( A ) ∧ M transitive and Cd-correct ∧ M | = ZFC −∗ ∧ A ∈ M ∧ M | = ( A | = L I φ ))iff ∀ M ((( ρ H ( M ) ≤ ρ H ( A ) ∧ M transitive and Cd-correct ∧ M | = ZFC −∗ ∧ A ∈ M ) → M | = ( A | = L I φ ))As in the proof of Lemma 4.9, we know that for any A we can let θ = | trcl( A ) | + ,so that | H θ | ≤ θ = 2 ρ H ( A ) + ≤ ρ H ( A ) and H θ is Cd-correct by Lemma 4.8 (1). Thus, the relation | = L I is ∆ F (Cd) forthe definably bounding function F ( α ) = 2 α .Now we check (2) of bounded symbiosis. Again, looking at the proof of Proposi-tion 3.5, we see that clauses (b) and (c) are L I -sentences, and (a) is Π B ( L I ) sincewe do not need to add new sorts. The only issue, then, is to prove that “( A, E )is well-founded” is Σ B ( L I ), which is less trivial because the method describedpreviously does not yield an upper bound on the size of the second sort. So weneed to adapt this method. The idea is to add a new linear ordering ( B, < ) tothe structure (
A, E ), and a function f : A → B , such that B plays the role ofthe appropriate cardinals ℵ α .Suppose ( B, < ) is a linear order. For b ∈ B let b ↓ = { b (cid:48) ∈ B : b (cid:48) < b } denotethe set of < -predecessors of b . We say that b is cardinal-like if for every b (cid:48) < b we have | b (cid:48) ↓| < | b ↓| , and the set B itself is cardinal-like if for every b ∈ B wehave | b ↓| < | B | .Consider the language with two sorts s and s , a binary relation symbol E in s , a binary relation < in s , and a function symbol f from s to s . In thefollowing proof, we will informally refer to the domains of the respective sortsby A and B .First define the following abbreviations:Inf( b ) ≡ ∀ B x < b I yz ( y < b ∧ y (cid:54) = x )( z < b )14.e., b has infinitely many < -predecessors.Like( b ) ≡ ∀ B b (cid:48) < b ¬ I yz ( y < b )( z < b (cid:48) )) , i.e., b is cardinal-like. Let Φ be the conjunction of the following L I -formulas:(i) < is a linear order;(ii) ∀ B b ¬ I xy ( x < b )( y = y )i.e., “ B is cardinal-like”;(iii) ∀ B b (Inf( b ) → ∃ B b (cid:48) ≤ b (I xy ( x < b )( y < b (cid:48) ) ∧ Like( b (cid:48) )))i.e., “no infinite cardinals are skipped”;(iv) ∀ A a ∀ A a (cid:48) ( aEa (cid:48) → f ( a ) < f ( a (cid:48) ))i.e., “ f is order-preserving”;(v) ∀ A a (Inf( f ( a )) ∧ Like( f ( a )))i.e., “every f ( a ) is infinite and cardinal-like”;(vi) ∀ A a ∀ B b ( b < f ( a ) → ∃ A a (cid:48) ( a (cid:48) Ea ∧ b ≤ f ( a (cid:48) ))i.e., “every | f ( a ) ↓| is the least cardinal higher than | f ( a (cid:48) ) ↓| for all a (cid:48) Ea ”;(vii) ∀ B b ∃ A a ( b ≤ f ( a ))i.e., “the range of f is cofinal in B ”.Now we prove several claims, which together imply that “( A, E ) is well-founded”is Σ B ( L I ). For ease of notation we will identify the symbols E, < and f withtheir respective interpretations. Claim 5.3. ( A, E ) is well-founded iff ( A, B, E, <, f ) | = L I Φ for some B, < and f .Proof. First, suppose (
A, E ) is well-founded. Let rk E be the rank functioninduced by E , let B = ℵ rk E ( A ) , and let f ( a ) = ℵ rk E ( a ) . Then it is easy to verifythat ( A, B, E, <, f ) | = L I Φ. Conversely, suppose (
A, B, E, <, f ) | = Φ. Then forevery aEa (cid:48) we have | f ( a ) ↓| < | f ( a (cid:48) ) ↓| , as follows easily from the fact that < istransitive, that f is order-preserving, and that every f ( a ) is cardinal-like. Butthen E must be well-founded. (Claim 5.3) Claim 5.4.
Suppose ( A, B, E, <, f ) | = L I Φ . Then1. For all b ∈ B and all λ < | b ↓| , there exists c < b such that λ ≤ | c ↓| < | b ↓| .2. For all b ∈ B and all λ < | b ↓| , there exists d < b such that | d ↓| = λ .3. For all λ < | B | , there exists d such that | d ↓| = λ .Proof.
15. Let b ∈ B and λ < | b ↓| . By (iii) there is a b (cid:48) < b such | b (cid:48) ↓| = | b ↓| and b (cid:48) iscardinal-like. We claim that there is c < b (cid:48) such that λ ≤ | c ↓| . Towardscontradiction, suppose this is false. Let { c α : α < | b (cid:48) ↓|} enumerate b (cid:48) ↓ and consider the initial λ -sequence of this enumeration, i.e., { c α : α <λ } . This sequence cannot be < -cofinal in b (cid:48) , otherwise we would have | b (cid:48) ↓| = | (cid:83) α<λ ( c α ↓ ) | ≤ λ × λ = λ , which is a contradiction. Therefore,there is c < b (cid:48) such that { c α : α < λ } ⊆ c ↓ . But then λ ≤ | c ↓| , also acontradiction.2. First apply (1) to find c < b such that λ ≤ | c ↓| < | b ↓| . Apply again tofind c < c such that λ ≤ | c ↓| < | c ↓| , etc. By well-foundedness, thisprocess will stop after finitely many steps and we will find d < b such that λ = | d ↓| .3. By an analogous argument as in (1) above, and using (ii), we first find b ∈ B such that λ ≤ | b ↓| < | B | . Then proceed as in (2). (Claim 5.4). Claim 5.5.
Suppose ( A, B, E, <, f ) | = L I Φ . Then | A ∪ B | ≤ ℵ rk E ( A ) .Proof. We prove, by induction on E , that for all a ∈ A : | f ( a ) ↓| ≤ ℵ rk E ( a ) . Suppose the above holds for all a (cid:48) Ea . Towards contradiction suppose | f ( a ) ↓| > ℵ rk E ( a ) . By Claim 5.4 (2), we can find d < f ( a ) such that | d ↓| = ℵ rk E ( a ) . By(vi), there exists a (cid:48) Ea such that d ≤ f ( a (cid:48) ). But this means that ℵ rk E ( a ) = | d ↓| ≤ | f ( a (cid:48) ) ↓| = ℵ rk E ( a (cid:48) ) which is a contradiction since rk E ( a (cid:48) ) < rk E ( a ). This competes the induction.Completing the proof requires repeating the above argument once more: if | B | > ℵ rk E ( A ) , then by Claim 5.4 (3) there is d ∈ B such that | d ↓| = ℵ rk E ( A ) ,and by (vii) there is a ∈ A with d ≤ f ( a ), implying ℵ rk E ( A ) = | d ↓| ≤ | f ( a ) ↓| = ℵ rk E ( a ) which is a contradiction since by definition rk E ( a ) < rk E ( A ). It follows that | A ∪ B | = | B | ≤ ℵ rk E ( A ) . (Claim 5.5) Proposition 5.6.
The pairs L and PwSt are bounded symbiotic.Proof.
A straightforward adaptation of the proof of Proposition 3.6 works. Us-ing the same trick as above, in (1) we see that | = L is ∆ F (PwSt) for F ( α ) = 2 α .For (2), we do not need to change anything since the class Q PwSt is already L -axiomatisable. 16 The Upwards Structural Reflection principle
Now we consider a reflection number analogous to the one in Definition 1.2which, as in [2], will allow us to connect the strength of existence of upwardL¨owenheim-Skolem numbers for strong logics to large cardinals.
Definition 6.1.
Let R be a Π predicate in the language of set theory. The bounded upwards structural reflection number USR ( R ) is the least κ such that:For every definably bounding function F , and every Σ F ( R )-definableclass of τ -structures in a fixed vocabulary τ closed under isomor-phisms:If there is A ∈ K with |A| ≥ κ , then for every κ (cid:48) > κ there is a B ∈ K with |B| ≥ κ (cid:48) and an elementary embedding e : A (cid:52) L ωω B .If there is no such cardinal, USR ( R ) is undefined. Remark 6.2.
In this definition, we are assuming that K is definable by aΣ F ( R )-formula without parameters . In particular, the definition presupposesthat τ , as a vocabulary, is itself Σ F ( R )-definable (e.g., finite). Notice that ifarbitrary τ were allowed, USR ( R ) would never be defined: for any κ , we couldtake a vocabulary τ with κ -many constant symbols and let K be the class of τ -structures such that every element is the interpretation of a constant symbol,which is ∆ id1 in τ . One could avoid this problem by considering classes definedwith parameters of a limited H -rank; but then, to prove results like the onesin this section, one would need to extend the corresponding logic in such a waythat the parameter can be defined. For the current paper, the parameter-freeversion will be sufficient.Our main theorem below is proved for logics which have ∆ -definable syntaxand dependence number ω . This is necessary if we want to avoid parameters—recall the discussion in Section 2. All logics obtained by adding finitely manyquantifiers to first- or second-order logic, such as L WF , L I , L and the examplesin [2, Proposition 4], are covered by this theorem. For the ULST -principle, seeDefinition 2.4 and recall that since we are assuming dep( L ∗ ) = ω , ULST ω ( L ∗ ) = ULST ∞ ( L ∗ ). Theorem 6.3 (Main Theorem) . Let L ∗ be a logic with ∆ -definable syntax and dep( L ∗ ) = ω , and let R be a Π predicate. Assume that L ∗ and R are boundedlysymbiotic. Then the following are equivalent: (1) ULST ∞ ( L ∗ ) = κ , (2) USR ( R ) = κ .Proof. (2) ⇒ (1). Suppose USR ( R ) = κ . Let φ be an L ∗ -formula, and let A | = L ∗ φ with |A| ≥ κ . Letting κ (cid:48) be any cardinal above κ , the goal is to finda super-structure B of A such that B | = L ∗ φ and |B| ≥ κ (cid:48) .Consider the class K = Mod( φ ). By condition (1) of bounded symbiosis , K is∆ F ( R )-definable, hence Σ F ( R ), with parameter φ . However, since dep( L ∗ ) = ω ,we may assume that the vocabulary of φ is finite. Moreover, L ∗ has a ∆ -definable syntax, so φ is ∆ -definable, therefore K is in fact Σ F ( R )-definable without parameters. It is also clearly closed under isomorphisms.17pplying USR ( R ) we find a B (cid:48) ∈ K , such that |B (cid:48) | ≥ κ (cid:48) and there is e : A (cid:52) L ωω B (cid:48) . We can also easily find B ∼ = B (cid:48) such that A is a substructure of B ,and this is what we need.(1) ⇒ (2). Now assume ULST ∞ ( L ∗ ) = κ , and let K be a Σ F ( R )-definabletransitive class of τ -structures, with Φ( x ) the defining Σ F ( R )-formula.Since the USR -principle involves elementary embeddings whereas
ULST doesnot, the proof must proceed indirectly. The intuition is that we first embed agiven structure
A ∈ K in a larger structure that includes a model of set theory H θ and includes Skolem functions for first-order existential sentences; then weapply ULST to (a further extension of) this larger structure. To make sure thatenlarging the set-theoretic structure also yields an enlargement of the originalstructure, we must carefully keep track of the relations between cardinalitiesgiven by the various bounds in the definition of bounded symbiosis . See Figure3. ℳℳ 𝒩𝒩 ( N , (cid:15464) ) 𝜋 proj. to 𝜏ʹ ⊆ 𝒜 𝒜 ʹ 𝜃 H { x : x E c } 𝒩 𝒩 proj. to 𝜏ʹ Figure 3: Structure of the proof.Assume that τ is in one sort (a similar proof works in the general case). Similarlyto the proof of Lemma 4.9, define a vocabulary τ (cid:48) with two sorts: s and s , withall of the symbols occurring in τ written in sort s . Let E be a binary relationsymbol in sort s , G a function symbol from s to s , and c a constant symbol insort s . In addition, for every quantifier-free first-order formula ψ ( x, y , . . . , y n )in the language E, c , add an n -ary function symbol f ψ of sort s .Let K ∗ be the class of all structures M := (cid:16) M, A, E M , G M , c M , { f M ψ } (cid:17) suchthat 18. ( M, E M ) | = ZFC −∗ ,2. ( M, E M ) ∈ Q R , i.e., it is isomorphic to a transitive R -correct model,3. | M | ≤ F ( | A | ) ,4. M | = Φ( c ), written with E instead of ∈ ,5. M | = ∀ ¯ z ( ∃ xψ ( x, ¯ z ) → ψ ( f ψ (¯ z ) , ¯ z )) for every quantifier-free ψ .6. M | = G is a bijection between A and { x : x E c } .Conditions 1, 4 and 6 are in first-order logic, whereas 2 is ∆ B ( L ∗ )-axiomatizableby the equivalent condition (2) of bounded symbiosis . Moreover, 3 is Σ B ( L ∗ )-axiomatisable by the definition of “definably bounding” (Definition 4.3), byExample 4.4, and the discussion following it. Finally, while 5 might look likean infinite set of sentences (and we are not assuming that L ∗ is infinitary), it isstill true that, since | = is ∆ id1 , the entire condition 5 can be expressed in a ∆ id1 way in set theory. By condition (2) of bounded symbiosis , the class of modelssatisfying 5 is ∆ B ( L ∗ ). Therefore, K ∗ is Σ B ( L ∗ ).Let A be a structure in K with |A| ≥ κ , and let κ (cid:48) > κ be any cardinal. Since K is closed under isomorphisms, we may assume wlog. that A is transitive. Ourgoal is to find A (cid:48) ∈ K such that |A (cid:48) | ≥ κ (cid:48) and A (cid:52) L ωω A (cid:48) .Let θ := F ( ρ H ( A )) + = F ( |A| ) + , choose Skolem functions f H θ ψ : H nθ → H θ , andconsider the structure M := ( H θ , A, ∈ , id , A , { f H ψ ψ } )Clearly M satisfies 1, 5 and 6 of the definition of K ∗ . Moreover, due to Lemma4.8 (1), H θ is R -correct and Φ( A ) is absolute for H θ . Hence, 2 and 4 are satisfiedas well. Finally, 3 holds because | H θ | ≤ θ = 2 F ( |A| ) + ≤ F ( |A| ) . Therefore
M ∈ K ∗ .Let χ be an L ∗ -sentence in an extended vocabulary τ (cid:48)(cid:48) such that K ∗ is a“bounded projection” of Mod( χ ). Let h : Ord → Ord be the function as inLemma 4.2.Let M = ( M , . . . ) be such that M | = χ and M (cid:22) τ (cid:48) = M . Since |M | ≥|M| ≥ |A| ≥ κ , we can apply ULST ∞ ( L ∗ ) = κ to find N such that N | = χ and |N | > h (cid:16) F ( κ (cid:48) ) (cid:17) , and M ⊆ N (i.e., M is a substructure of N ). Let N = N (cid:22) τ (cid:48) . We write N = ( N, B, E N , G N , c N , { f N ψ } ) for this structure.Let ( ¯ N , ∈ ) be the transitive collapse of ( N, E N ), and c N be the image of c N under this collapse. We claim that A (cid:48) := c N is the model we are looking for. Claim 6.4. A (cid:48) ∈ K .Proof. Since
N ∈ K ∗ , we know that ( N, E N ) | = Φ( c ) (written in E ), andtherefore ¯ N | = Φ(¯ c ) (written in ∈ ). Also, since ( N, E N ) ∈ Q R , we know that¯ N is R -correct, in particular, Σ ( R ) formulas are upwards absolute. ThereforeΦ( A (cid:48) ) is true, so A (cid:48) ∈ K . 19 laim 6.5. κ (cid:48) < |A (cid:48) | .Proof. By the definition of the function h as in Lemma 4.2, we know that |N | ≤ h ( |N | ). Thus we have h (cid:16) F ( κ (cid:48) ) (cid:17) < |N | ≤ h ( |N | )and since h is order-preserving, 2 F ( κ (cid:48) ) < |N | . By condition 3 of the definitionof K ∗ , we have | N | ≤ F ( | B | ) . Therefore 2 F ( κ (cid:48) ) < | N | ≤ F ( | B | ) , from whichit follows that κ (cid:48) < | B | . Finally, by condition 6 we get that | B | = |{ x ∈ N : xE N c N | = |A (cid:48) | . Claim 6.6.
There is an L ωω -elementary embedding from A to A (cid:48) .Proof. By condition 5, both models M and N satisfy the axioms for Skolemfunctions concerning first-order quantifier-free formulas in { E, c } . In addition,since M is a substructure of N , the interpretations of f ψ coincide between themodels, i.e., f N ψ (cid:22) H θ = f H θ ψ for every ψ . Thus, if N | = ∃ xψ ( x, ¯ z ) and ¯ z ∈ H θ ,then N | = ψ ( f ψ (¯ z ) , ¯ z ), so ( H θ , ∈ , A ) | = ψ ( f ψ (¯ z ) , ¯ z ). It follows that N and( H θ , ∈ , A ) satisfy the same Σ -formulas in { E, c } .Let π : N → ¯ N be the collapsing map. Since the first-order satisfaction relationis ∆ , for every first-order φ and for every ¯ a = a , . . . a n ∈ A we have A | = φ (¯ a ) ⇔ H θ | = ( A | = φ (¯ a )) ⇔ N | = ( c | = φ (¯ a )) ⇔ ( ¯ N , ∈ , A (cid:48) ) | = ( c | = φ ( π (¯ a ))) ⇔ A (cid:48) | = φ ( π (¯ a ))).Hence π (cid:22) A : A (cid:52) L ωω A (cid:48) as required. PwSt and second order logic
In this section we apply our results to determine upper and lower bounds for thelarge cardinal strength of
USR (PwSt) and
ULST ∞ ( L ), which will also yieldupper bounds for other symbiotic pairs L ∗ and R . The main point is that PwStcan be seen as an upper bound for all Π predicates. The following is not hardto verify (see [5, Section 6.5] for the details). Fact 7.1.
The function α (cid:55)→ V α is Σ F (PwSt) -definable (for suitable F ). Also,the function H that maps every infinite successor cardinal θ to H θ is Σ F (PwSt) -definable. Lemma 7.2.
For every Π predicate R , if φ is Σ F ( R ) then it is Σ F (PwSt) . In this proof we have occasionally identified the syntax and semantics of first-order logicfor ease of readability. roof. Suppose φ is Σ F ( R ). Then for every a we have φ ( a ) ⇔ ∃ H θ (cid:16) ρ H ( H θ ) < F ( ρ H ( a )) ∧ H θ | = φ ( a ) (cid:17) . By the previous fact, “being H θ ” is Σ F (cid:48) (PwSt)-definable (possibly another F (cid:48) ).In conjunction with the upper bound, the whole expression is also Σ F (cid:48)(cid:48) (PwSt)-definable (for F (cid:48)(cid:48) being the maximum of F (cid:48) and α (cid:55)→ F ( α ) ). To see that theequivalence holds, let θ = F ( ρ H ( a )) + . Then ρ H ( H θ ) ≤ θ ≤ F ( ρ H ( a )) , andmoreover φ ( a ) is absolute for H θ by Lemma 4.8 (2). Corollary 7.3.
1. For every Π predicate R we have USR ( R ) ≤ USR (PwSt) . In particular,if USR (PwSt) is defined then
USR ( R ) is also defined.2. If L ∗ is any logic which is boundedly symbiotic to some Π -predicate R ,has ∆ -definable syntax and dep( L ∗ ) = ω , then ULST ∞ ( L ∗ ) ≤ ULST ∞ ( L ) .In particular, if ULST ∞ ( L ) is defined, then so is ULST ∞ ( L ∗ ) . A famous result of Magidor [8] shows that the least cardinal κ for which L satisfies a κ -version of compactness, is the least extendible cardinal. One canshow that this version of compactness implies ULST ∞ ( L ) = κ . Therefore anextendible cardinal provides an upper bound for ULST ∞ ( L ) and USR (PwSt),as well as other pairs L ∗ and R satisfying bounded symbiosis and the conditionsof Theorem 6.3. For completeness, we include a short proof of this fact. Theorem 7.4 (Magidor [8]) . If κ is an extendible cardinal, then USR (PwSt) =
ULST ( L ) ≤ κ. Moreover,
USR ( R ) ≤ κ for every Π predicate R , and ULST ∞ ( L ∗ ) ≤ κ forany L ∗ which is boundedly symbiotic with some Π predicate, and which has ∆ -definable syntax and dep( L ∗ ) = ω .Proof. Let κ be extendible, and we will prove that USR (PwSt) ≤ κ . The otherstatements follow by Theorem 6.3 and Corollary 7.3.Let K be a Σ F (PwSt)-definable class of τ -structures closed under isomorphisms.Fix some A ∈ K with |A| ≥ κ . Let κ (cid:48) > κ be arbitrary. Let η > κ (cid:48) be such that A ∈ V η and V η | = Φ( A ) ∧ ( |A| ≥ κ ). Then there is an elementary embedding J : V η (cid:52) V θ for some θ , and J ( κ ) > η > κ (cid:48) . But then by elementarity we have V θ | = Φ( J ( A )) ∧ | J ( A ) | ≥ J ( κ ). Since V θ is PwSt-correct, Φ( J ( A )) holds, so J ( A ) ∈ K . Moreover, since θ is sufficiently large, we have |A| ≥ J ( κ ) > η >κ (cid:48) . Finally, A (cid:52) L ωω J ( A ) holds by elementarity and first-order definability of“ A | = φ ”.Now we look at how much large cardinal strength we can obtain from USR (PwSt).
Theorem 7.5. If USR (PwSt) is defined, then there exists an n -extendiblecardinal for every natural number n > . roof. Assume that
USR (PwSt) = κ . Let K to be the class of all structureswhich are isomorphic to ( V α + n , ∈ , α ), in the language { E, a } .By Fact 7.1, being a structure of the form ( V α + n , ∈ , α ) is Σ F (PwSt)-definable.Then ( M, a, E ) ∈ K iff ∃ ( V α + n , ∈ , α ) and ∃ f : ( M, a, E ) ∼ = ( V α + n , ∈ , α ). Noticethat also ρ H ( V α + n ) ≤ ρ H ( M ). Thus, K is Σ F (PwSt)-definable.Take any µ ≥ κ . Since ( V µ + n , µ, ∈ ) ∈ K , by USR (PwSt) there exists anelementary embedding J : ( V µ + n , µ, ∈ ) (cid:52) L ωω ( V β + n , β, ∈ )for some β > µ , which maps µ to β . Let λ be the critical point of J , which is ≤ µ . But then J (cid:22) V λ + n : V λ + n (cid:52) V J ( λ )+ n (this includes the case λ = µ ), since: V λ + n | = ϕ ( x , . . . , x n ) ⇔ V µ + n | = ( V λ + n | = ϕ ( x , . . . , x n )) ⇔ V β + n | = ( J ( V λ + n ) | = ϕ ( J ( x ) , . . . , J ( x n ))) ⇔ V β + n | = ( V J ( λ )+ n | = ϕ ( J ( x ) , . . . , J ( x n ))) ⇔ V J ( λ )+ n | = ϕ ( J ( x ) , . . . , J ( x n ))Since n < J ( λ ), it follows that λ is n -extendible. Corollary 7.6. If ULST ∞ ( L ) is defined then there exists an n -extendible car-dinal for every n . Notice that the only reason that the proof works for n < ω and not arbitrary α , is because the class K needs to be definable. It is easy to adapt the proofto show that there exists a γ -extendible cardinal for any Σ F (PwSt)-definableordinal γ . In fact, we conjecture that the consistency strength is exactly anextendible. Conjecture 7.7.
USR (PwSt) and
ULST ∞ ( L ) are defined if and only if thereexists and extendible cardinal. The biggest question left open in this paper is the exact consistency strength of
USR (PwSt) and
ULST ∞ ( L ), i.e., Conjecture 7.7.Other questions that we have not investigated involve a similar analysis ofthe large cardinal strength for other symbiotic pairs L ∗ and R . Question 8.1.
What is the large cardinal strength (or, at least, lower andupper bounds), for the principles
USR ( R ) and ULST ∞ ( L ∗ ) , for other boundedlysymbiotic pairs R and L ∗ , such as the ones in [2, Proposition 4]? Another important issue, which we have not investigated in this paper, isthe study of various compactness properties of strong logics.
Definition 8.2.
A logic L ∗ is ( κ, γ )-compact if for every set T of sentences ofsize γ , if every < κ -sized subset of T has a model, then T has a model. A logic L ∗ is ( κ, ∞ )-compact if it is ( κ, γ )-compact for every γ . Classical compactnessis ( ω, ∞ )-compactness. 22ost strong logics are not ( ω, ∞ )-compact but may be ( κ, ∞ )-compact forsome κ . Often such a κ will exhibit large cardinal properties, e.g., Magidor’sresult on L [8]. As we mentioned in the previous section, it is easy to see that:If L ∗ is ( κ, ∞ )-compact then ULST ∞ ( L ∗ ) ≤ κ .We do not know whether the converse holds. The following questions seeminteresting and worth investigating: Question 8.3.
Assume that κ is a regular cardinal. For which logics does ULST ∞ ( L ∗ ) ≤ κ imply ( κ, ∞ ) -compactness? One can try to look for a set-theoretic principle involving ∆ ( R ) definableclasses of structures, which would correspond to ( κ, ∞ )-compactness in a similarway as in Theorem 6.3. Question 8.4.
Is there a set-theoretic principle P ( R ) , for classes definableusing a Π -parameter R , such that if R and L ∗ are (bounded) symbiotic, then P ( R ) = κ if and only if L ∗ is ( κ, ∞ ) -compact? Answering the last question could involve extensions of partial orders withina fixed ∆ ( R )-class, using ideas from [10]. Notice, however, that when dealingwith compactness, large vocabularies are essential, so the corresponding princi-ples will require the use of parameters, which will restrict the class of logics L ∗ to which it can apply. Acknowledgements:
We would like to thank Soroush Rafiee Rad and RobertPassmann for initiating this research and providing valuable input.
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